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In various contexts in mathematical physics, such as out-of-equilibrium physics and the asymptotic information theory of many-body quantum systems, one needs to compute the logarithm of a positive unbounded operator. Examples include the von Neumann entropy of a density matrix and the flow of operators with the modular Hamiltonian in the Tomita-Takesaki theory. Often, one encounters the situation where the operator under consideration, which we denote by ∆, can be related by a perturbative series to another operator ∆0, whose logarithm is known. We set up a perturbation theory for the logarithm log ∆. It turns out that the terms in the series possess a remarkable algebraic structure, which enables us to write them in the form of nested commutators plus some "contact terms". [ABSTRACT FROM AUTHOR]

The small parameter method allows one to construct solutions of differential equations in the form of power series and has become widespread in mathematical physics. In most cases, these series are asymptotically convergent. The aim of this work is to find conditions for the ordinary convergence of series in powers of a small parameter representing solutions of perturbation theory problems. [ABSTRACT FROM AUTHOR]

Predicting the electronic structure and properties of molecular systems that display strong electron correlation effects continues to remain a fundamental theoretical challenge. This perspective discusses the recent progress and current challenges in multireference wave function methods for dynamical electron correlation, focusing on systematically improvable methods that go beyond the limitations of configuration interaction and perturbation theory. [ABSTRACT FROM AUTHOR]

A benchmark set of relevant geometries of a model protein, the N-acetylphenylalanylamide, is presented to assess the validity of the approximate second-order coupled cluster (CC2) method in studying low-lying excited states of such bio-relevant systems. The studies comprise investigations of basis-set dependence as well as comparison with two multireference methods, the multistate complete active space 2nd order perturbation theory (MS-CASPT2) and the multireference difference dedicated configuration interaction (DDCI) methods. First of all, the applicability and the accuracy of the quasi-linear multireference difference dedicated configuration interaction method have been demonstrated on bio-relevant systems by comparison with the results obtained by the standard MS-CASPT2. Second, both the nature and excitation energy of the first low-lying excited state obtained at the CC2 level are very close to the Davidson corrected CAS+DDCI ones, the mean absolute deviation on the excitation energy being equal to 0.1 eV with a maximum of less than 0.2 eV. Finally, for the following low-lying excited states, if the nature is always well reproduced at the CC2 level, the differences on excitation energies become more important and can depend on the geometry. [ABSTRACT FROM AUTHOR]

By appealing to the coupling parameter series expansion to calculate the first seven perturbation coefficients of the high temperature series expansion (HTSE) of the free energy, analysis of convergence and low temperature adaptability of the HTSE in calculating fluid thermodynamic properties is performed for the first time; the fluid thermodynamic properties considered include critical parameters, vapor-liquid coexistence curve, thermodynamic characteristic functions, chemical potential, pressure, and constant volume excess heat capacity. To proceed with the analysis, a well known square well model is used as sample; the well widths considered range over a wide interval, and the relevant temperatures amenable to simulation calculations (used as 'exact' results to analyze the HTSE) can be both very high and very low. The main discoveries reached are summarized as follows: (1) The HTSE usually converges at the 4th-order truncation, but with decrease of the temperature considered, the lowest truncation order, which makes the HTSE to converge, tends to rise. As a conservative estimate, it is considered that the HTSE always converges for reduced temperature T* higher than 0.25, whereas for T* < 0.25 there appear signs indicating that the HTSE may diverge from the 7th-order truncation. (2) Within the temperature interval with T* >= 0.5, the HTSE converges approximately to the correct solution, and the HTSE can be reliably used to calculate the fluid thermodynamic properties, and within this temperature interval, the 4th-order truncation is enough; whereas for T* < 0.5, such as within the temperature interval with 0.275 <= T* <= 0.355, although the HTSE does converge, it does not converge to the correct solution, and the deviations between the HTSE calculations and MC simulations become an ever-prominent issue with the rising of the density, and the slopes of the thermodynamic properties over density are not satisfactorily represented. As a result, the HTSE is not suited for calculations for temperature interval T* < 0.5. [ABSTRACT FROM AUTHOR]

The integrodifferential equations of mathematical physics are objects of research, and construction of interpolating polynomials to obtain approximate solutions of such equations is the subject of investigation. This paper lays out a technique for constructing approximate expressions for functionals on solutions of integrodifferential equations which are an analog of the Hermite polynomial used to interpolate functions. In the example of the diffusion equation, it is shown that the use of such basis solutions allows a substantial increase in the accuracy of the approximate representation of the functionals in comparison to the first approximation of perturbation theory with practically the same computational costs. [ABSTRACT FROM AUTHOR]

The equations governing the perturbations of the Schwarzschild metric satisfy the Regge–Wheeler–Zerilli–Moncrief system. Applying the technique introduced in Andersson and Blue (Ann Math 182(2):787–853, 2015), we prove an integrated local energy decay estimate for both the Regge–Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the r p hierarchy estimates (Dafermos and Rodnianski, in: Exner (ed) XVIth international congress on mathematical physics, World Scientic, London, pp 421–433, 2009; Schlue in Anal PDE 6:515–600, 2013) to prove that both the Regge–Wheeler and Zerilli variables decay as t - 3 2 in fixed regions of r. [ABSTRACT FROM AUTHOR]

We have studied an iterative perturbative approach to optimize Jastrow factors in quantum Monte Carlo calculations. For an initial guess of the Jastrow factor we construct a corresponding model Hamiltonian and solve a first-order perturbation equation in order to obtain an improved Jastrow factor. This process is repeated until convergence. Two different types of model Hamiltonians have been studied for both energy and variance minimization. Our approach can be considered as an alternative to Newton’s method. Test calculations revealed the same fast convergence as for Newton’s method sufficiently close to the minimum. However, for a poor initial guess of the Jastrow factor, the perturbative approach is considerably more robust especially for variance minimization. Usually only two iterations are sufficient in order to achieve convergence within the statistical error. This is demonstrated for energy and variance minimization for the first row atoms and some small molecules. Furthermore, our perturbation analysis provides new insight into some recently proposed modifications of Newton’s method for energy minimization. A peculiar feature of the analysis is the continuous use of cumulants which guarantees size-consistency and provides least statistical fluctuations in the Monte Carlo implementation. [ABSTRACT FROM AUTHOR]

We present molecular applications of a spin free size-extensive state-specific multireference perturbation theory (SS-MRPT), which is valid for model functions of arbitrary spin and generality. In addition to the singlet states, this method is equally capable to handle nonsinglet states. The formulation based on Rayleigh–Schrödinger approach works with a complete active space and treats each of the model space functions democratically. The method is capable of handling varying degrees of quasidegeneracy and of ensuring size consistency as a consequence of size extensivity. In this paper, we illustrate the effectiveness of the Mo\ller–Plesset (MP) partitioning based spin free SS-MRPT [termed as SS-MRPT(MP)] in computations of energetics of the nonsinglet states of several chemically interesting and demanding molecular examples such as LiH, NH2, and CH3. The spectroscopic constants of 3Σ- state of NH and OH+ molecular systems and the ground 1Σg+ as well as excited 3Σu+ states of N2 have been investigated and comparison with experimental and full configuration interaction values (wherever available) has also been provided. We have been able to demonstrate here that the SS-MRPT(MP) method is an intrinsically consistent and promising approach to compute reliable energies of nonsinglet states over different geometries. [ABSTRACT FROM AUTHOR]

Adding a fraction of the nonlocal exchange operator to the local orbital-dependent exchange potential improves the many-body perturbation expansion based on the Kohn–Sham determinant. The effect of such a hybrid scheme on the performance of the orbital-dependent correlation functional from the second-order perturbation theory (PT2H) is investigated numerically. A small fraction of the nonlocal exchange is often sufficient to ensure the existence of the self-consistent solution for the PT2H potential. In the He and Be atoms, including 37% of the nonlocal exchange leads to the correlation energies and electronic densities that are very close to the exact ones. In molecules, varying the fraction of the nonlocal exchange may result in the PT2H energy closely reproducing the CCSD(T) value; however such a fraction depends on the system and does not always result in an accurate electronic density. We also numerically verify that the “semicanonical” perturbation series includes most of the beneficial effects of the nonlocal exchange without sacrificing the locality of the exchange potential. [ABSTRACT FROM AUTHOR]

A new explicitly correlated local MP2-F12 method is proposed in which the error caused by truncating the virtual orbital space to pair-specific local domains is almost entirely removed. This is achieved by a simple modification of the ansatz for the explicitly correlated wave function, which makes it possible that the explicitly correlated terms correct both for the basis set incompleteness error as well as for the domain error in the LMP2. Benchmark calculations are presented for 21 molecules and 16 chemical reactions. The results demonstrate that the local approximations have hardly any effect on the accuracy of the computed correlation energies and reaction energies, and the LMP2-F12 reaction energies agree within 0.1–0.2 kcal/mol with estimated MP2 basis set limits. [ABSTRACT FROM AUTHOR]

We present a theoretical scheme for a semiquantitative analysis of electronic structures of magnetic transition metal dimer complexes within spin density functional theory (DFT). Based on the spin polarization perturbational orbital theory [D.-K. Seo, J. Chem. Phys. 125, 154105 (2006)], explicit spin-dependent expressions of the spin orbital energies and coefficients are derived, which allows to understand how spin orbitals form and change their energies and shapes when two magnetic sites are coupled either ferromagnetically or antiferromagnetically. Upon employment of the concept of magnetic orbitals in the active-electron approximation, a general mathematical formula is obtained for the magnetic coupling constant J from the analytical expression for the electronic energy difference between low-spin broken-symmetry and high-spin states. The origin of the potential exchange and kinetic exchange terms based on the one-electron picture is also elucidated. In addition, we provide a general account of the DFT analysis of the magnetic exchange interactions in compounds for which the active-electron approximation is not appropriate. [ABSTRACT FROM AUTHOR]

The application of the frozen-core approximation (FCA) and effective core potentials (ECPs) within symmetry-adapted perturbation theory (SAPT) has been investigated and implemented. Unlike in the case of conventional electronic-structure theories, the development of a frozen-core version of SAPT is not straightforward. In particular, the FCA realizations neglecting excitations from core orbitals and restricting all summation indices to valence orbitals only are no longer equivalent. It is shown that it is necessary in SAPT to keep some terms containing products of the valence orbitals of one monomer and the core orbitals of the other one in the exchange-energy components. When these terms are included or, equivalently, the “infinite-excitation-energy” approximation omitting only the excitations from the core orbitals is used, the accuracy of the frozen-core approximation in SAPT matches that obtained in supermolecular perturbational and coupled-cluster methods. If these terms are neglected, i.e., within the “index-range-restriction” approximation, several exchange corrections are significantly underestimated. When ECPs are used in SAPT, the accuracy of the interaction energies is as good as in conventional supermolecular methods, provided that the residual supermolecular Hartree-Fock term is included. We have found that only some types of ECPs can be reliably used for calculations of interaction energies both in SAPT and in supermolecular approaches. For systems containing heavy atoms, both FCA and the use of ECPs lead to very significant savings of computer time. [ABSTRACT FROM AUTHOR]

The existence of a seam of conical intersection, the reference seam, does not rule out the existence of additional disjoint seams of conical intersection. These disjoint seams intersect the g-h planes of the reference seam, a region usually assumed to be devoid of intersections, potentially leading to unexpected points of degeneracy in close proximity to the original conical intersection. Here the authors show how the locus of these disjoint seams can be predicted employing a Hamiltonian derived from second-order perturbation theory. Dramatic differences between the g-h planes of the reference and disjoint seams are found and are expected to have a profound impact on nuclear dynamics. Numerical studies of both high symmetry (D3h, C3H3) and low symmetry (C2v, C2H2N) species are presented. [ABSTRACT FROM AUTHOR]

Theoretical analysis based on a nondegenerate perturbation method is presented for two-dimensional buckling patterns of a thin metal-capped polymer bilayer on a rigid substrate. The analysis reveals that isotropic one-dimensional buckling mode has the lowest free energy, which contrasts previous work on the buckling of a thin metal film on the semi-infinitely thick polymer layer studied by Chen and Hutchinson [J. Appl. Mech. 71, 597 (2004)]. In the low stress region, however, checkerboard buckling mode has the lowest energy. Herringbone buckling mode is shown to be represented by a linear combination of the checkerboard mode and the isotropic one-dimensional mode. Disorder parameter values are given for these various two-dimensional buckling modes on the basis of an envelope function. [ABSTRACT FROM AUTHOR]

A systematic study of the deviation from size consistency of the multireference second-order Møller-Plesset perturbation theory (MRMP2) method is presented. The size-consistency error is shown to depend on the number of monomers in a supermolecule calculation, size of basis set, number of correlated valence electrons, and size of active space. HF, F2, and N2 are used as test cases, with stretched bonds, to include simple, well-defined multireference character. This is essential in ensuring that MRMP2 is being tested as a multireference method. It is concluded that the MRMP2 and other multireference perturbation theory methods can exhibit significant size-consistency errors, and that the size of the error depends on the manner in which the perturbation theory is implemented. © 2005 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A perturbative approach is employed to solve the Bloch-Torrey equations in the presence of distant-dipole fields in nuclear magnetic resonance. The procedure, although only carried out to first order in the perturbation parameter a=1/k2Dτd, could, in principle, be generalized to higher orders. Here D is the diffusivity, τd the dipolar demagnetization time, and k is the wave vector of the spatial modulation of magnetization produced by the magnetic field gradient. The results are especially interesting for dilute binary mixtures consisting of molecular species with different diffusivities. In this case the calculated two-dimensional correlation spectroscopy revamped by asymmetric Z-gradient echo detection spectra are shown to be free from some inadequacies resulting from a simplistic application of standard approximations. © 2005 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We describe a modification of the inverted perturbation approach method allowing to construct physically sensible potential energy curves for electronic states of diatomic molecules even when some parts of the potential are not adequately characterized by the experimental data. The method is based on a simple regularization procedure, imposing an additional constraint on the constructed potential curve. In the present work it is applied to the double minimum 4 1Σu+ state of Na2, observed experimentally by polarization labeling spectroscopy technique.© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The interactions of Ga(2P:4s24p1, 2S:4s25s1, and 2P:4s25p1) with SiH4 are studied by means of Hartree–Fock self-consistent field (SCF) and multiconfigurational SCF followed by extensive variational and perturbational second-order multireference Møller–Plesset configuration by perturbation selected by iterative process calculations, using relativistic effective core potentials. The Ga atom in its 2P(4s25p1) state can spontaneously insert into the SiH4. The Ga atom in its 2S(4s25s1) state is inserted into the SiH4. In this interaction the 3 2A′ potential energy surface initially attractive becomes repulsive after meeting the 2 2A′ surface linked with the Ga(2P:4s24p1)+SiH4 fragments. The two 2A′ curves (2 2A′ and X 2A′) derived from the interaction of Ga(2P:4s24p1) atom with silane molecule are initially repulsive. The 2 2A′ curve after an avoided crossing with the 3 2A′ curve goes down until it meets the X 2A′ curve. The 2 2A′ curve becomes repulsive after the avoided crossing with the X 2A′ curve. The X 2A′ curve becomes attractive only after its avoided crossing with the 2 2A′ curve. The lowest-lying X 2A′ potential leads to the HGaSiH3X 2A′ intermediate molecule. This intermediate molecule, diabatically correlated with the Ga(2S:4s25s1)+SiH4 fragments, which lies 1.5 kcal/mol above the ground state reactants leads to the GaH+SiH3 or H+GaSiH3 products through the dissociation channels. These products are reached from the HGaSiH3 intermediate without activation barriers. This work shows that the Ga atom at its first excited state in the presence of silane molecules in gas phase leads to the formation of SiH3 radicals, H atoms, GaH hydrides, as well as gallium silicide molecules. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We describe a modification of the inverted perturbation approach method allowing to construct physically sensible potential energy curves for electronic states of diatomic molecules even when some parts of the potential are not adequately characterized by the experimental data. The method is based on a simple regularization procedure, imposing an additional constraint on the constructed potential curve. In the present work it is applied to the double minimum 4 1Σu+ state of Na2, observed experimentally by polarization labeling spectroscopy technique.© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Incoherence in the controlled Hamiltonian is an important limitation on the precision of coherent control in quantum information processing. Incoherence can typically be modeled as a distribution of unitary processes arising from slowly varying experimental parameters. We show how it introduces artifacts in quantum process tomography and we explain how the resulting estimate of the superoperator may not be completely positive. We then go on to attack the inverse problem of extracting an effective distribution of unitaries that characterizes the incoherence via a perturbation theory analysis of the superoperator eigenvalue spectra. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An efficient method to compute analytical energy derivatives for local second-order Møller–Plesset perturbation energy is presented. Density fitting approximations are employed for all 4-index integrals and their derivatives. Using local fitting approximations, quadratic scaling with molecular size and cubic scaling with basis set size for a given molecule is achieved. The density fitting approximations have a negligible effect on the accuracy of optimized equilibrium structures or computed energy differences. The method can be applied to much larger molecules and basis sets than any previous second-order Møller–Plesset gradient program. The efficiency and accuracy of the method is demonstrated for a number of organic molecules as well as for molecular clusters. Examples of geometry optimizations for molecules with 100 atoms and over 2000 basis functions without symmetry are presented. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The potential energy surfaces have been constructed for the 1A′, 3A′, and 3A″ states of HBO by using the multireference perturbation theory with the basis set cc-pVTZ (6d,10f ). Two stationary points and a transition state have been characterized on all the three surfaces, which are in good agreement with available experiments and previous calculations. The interconversion pathways from metastable boron hydroxide BOH to the considerably more stable HBO are expounded based on the nature of the surfaces. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A full dimensional vibrational treatment of CHBr3 and CDBr3 using Van Vleck perturbation theory followed by a variational calculation is presented. The calculation of a force field, and its adjustment for better match with experiment, is discussed. The computed eigenstates and spectral features are compared to experiment. Changes in intensities of the ν1 and 2ν4 bands upon simple alterations of the dipole moment expansion are described. [ABSTRACT FROM AUTHOR]

A full dimensional vibrational treatment of CHBr3 and CDBr3 using Van Vleck perturbation theory followed by a variational calculation is presented. The calculation of a force field, and its adjustment for better match with experiment, is discussed. The computed eigenstates and spectral features are compared to experiment. Changes in intensities of the ν1 and 2ν4 bands upon simple alterations of the dipole moment expansion are described. [ABSTRACT FROM AUTHOR]

The geometries, the harmonic vibrational frequencies, and the Renner–Teller parameter have been reported for the NCO+(X 3Σ-), NCO(X 2Π,Ã 2Σ+,B 2Π,2 2Σ+), NCO-(X 1Σ+), CNO+(X), CNO(X 2Π,Ã 2Σ+,B 2Π,2 2Σ+), and CNO-(X 1Σ+) systems at the full valence–complete active space self-consistent-field (fv–CASSCF) level of theory. The 2Π electronic states of the NCO and CNO radicals have two distinct real vibrational frequencies for the bending modes and these states are subject to the type A Renner–Teller effect. The total energy of CNO+ without zero point energy correction of the linear geometry is ∼31 cm-1 higher than the bent geometry at the fv–CASSCF level and the inversion barrier vanishes after the zero point energy correction; therefore, the ground state of the CNO+ may possess a quasilinear geometry. The spin–orbit coupling constants estimated using atomic mean field Hamiltonian at the fv–CASSCF level of theory are in better agreement with the experimental values. The excitation energies, the electron affinity, and the ionization potential have been computed at the complete active space second order perturbation theory (CASPT2) and the multireference singles and doubles configuration (MRSD–CI) levels of theory. The computed values of the electric hyperfine coupling constants for the 14N atom in the ground state of the NCO radical agree well with the experimental data. The magnetic hyperfine coupling constants (HFCC’s) have been estimated employing the configuration selected MRSD–CI and the multireference singles configuration interaction (MRS–CI) methods using iterative natural orbitals (ino) as one particle basis. Sufficiently accurate value of the isotropic contribution to the HFCC’s can be obtained using an MRS–CI–ino procedure. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The perturbation method is developed to investigate the effective nonlinear dielectric response of Kerr composites when the external ac and dc electric field is applied. Under the external ac and dc electric field E[sub app]=E[sub a](1+sin ωt), the effective coupling nonlinear response can be induced by the cubic nonlinearity of Kerr nonlinear materials at the zero frequency, the finite basic frequency ω, the second and the third harmonics, 2ω and 3ω, and so on. As an example, we have investigated the cylindrical inclusions randomly embedded in a host and derived the formulas of the effective nonlinear dielectric response at harmonics in dilute limit. For a higher concentration of inclusions, we have proposed a nonlinear effective-medium approximation by introducing the general effective nonlinear response. With the relationships between the effective nonlinear response at harmonics and the general effective nonlinear response, we have derived a set of formulas of the effective nonlinear dielectric responses at harmonics for a larger volume fraction. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

• The physics adjoint and the mathematical adjoint are compared based on the 2-D/1-D solver. • Different adjoint flux functions are used to perform S&U analysis for comparison. • The SF96 problem and the PB-2 assembly problem are selected for verification. Sensitivity and Uncertainty analysis (S&U) based on the Classical Perturbation Theory (CPT) requires the solution of the adjoint flux. The adjoint flux is used as a weight function for sensitivity calculation. In the previous study, the MOC-based adjoint flux in the 2-D/1-D transport solver was given and applied to S&U analysis in the UAM benchmarks. The eigenvalue k obtained by the forward calculation is different from that obtained by the adjoint calculation. In order to illustrate the phenomenon, three different adjoint flux functions, the MOC physics adjoint, the CMFD physics adjoint, and the CMFD mathematical adjoint are solved based on the 2-D/1-D solver. Next, three different adjoint flux functions are used to perform S&U analysis. The SF96 problem and the PB-2 assembly problem are selected for verification. Recommendation is made for the proper application of the CMFD mathematical adjoint flux to sensitivity calculation due to less computation time. [ABSTRACT FROM AUTHOR]

We study the initial value problem for the half Ginzburg-Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation. [ABSTRACT FROM AUTHOR]

In previous work [13] we introduced a new box dimension method for computation of the number of limit cycles in planar slow-fast systems, Hausdorffi close to balanced canard cycles with one breaking mechanism (the Hopf breaking mechanism or the jump breaking mechanism). This geometric approach consists of a simple iteration method for finding one orbit of the socalled slow relation function and of the calculation of the box dimension of that orbit. Then we read the cyclicity of the balanced canard cycles from the box dimension. The purpose of the present paper is twofold. First, we generalize the box dimension method to canard cycles with two breaking mechanisms. Second, we apply the method from [13] and our generalized method to a number of interesting examples of canard cycles with one breaking mechanism and with two breaking mechanisms respectively. [ABSTRACT FROM AUTHOR]

Highlights • We have presented a new class of models of accelerating universe with gravitational coupling G (t) and cosmological term Λ(t) in the framework of general relativity with conharmonically flat space. • Our derived models are of great importance in the sense that the nature of decaying vacuum energy density Λ(t) is supported by recent cosmological observations. • The evolution of the universe approaches to ΛCDM model (r = 1 , s = 0). So, from the Statefinder parameter { r, s }, the behaviour of different stages of the evolution of the universe have been generated. • The background solution is stable against the perturbation of the graviton field. Abstract General exact solutions of the Einstein’s field equations by using conharmonically flat space with variable gravitational and cosmological “constants” for a spatially homogeneous and isotropic Friedmann–Robertson–Walker (FRW) model filled with perfect fluid has been obtained. To study the transit behaviour of Universe, we consider a law of variation of scale factor a (t) = (t k e t) 1 n which yields a time-dependent deceleration parameter (DP) q = − 1 + n k (k + t) 2 , comprising a class of models that depicts a transition of the universe from the early decelerated phase to the recent accelerating phase. We find that the time-dependent DP is reasonable for the present day Universe and give an appropriate description of the evolution of the universe. For n = 0.27 k , we obtain q 0 = − 0.73 which is similar to observed value of DP at present epoch. It is also observed that for n ≥ 2 and k = 1 , we obtain a class of transit models of the universe from early decelerating to present accelerating phase. For k = 0 , the universe has non-singular origin. In these models, we arrive at the decision that, from the structure of the field equations, the behaviour of Λ and G are related. Taking into consideration the observational data, we conclude that the Λ behaves as a positive decreasing function of time whereas G is increasing and tend to a constant value at the late time. Our derived model is in good agreement with ΛCDM model. Some physical and geometric properties of the models are also discussed. [ABSTRACT FROM AUTHOR]

Several different techniques via the improved sub-ODE method, the new mapping method, the new extended auxiliary equation method, the ( G ′/ G )-expansion method and the modified simple equation method are employed in this paper for finding solitons and other solutions to the improved perturbed nonlinear Schrodinger equation with the dual-power law nonlinearity. Comparing our new results with the well-known results are given. Comparing the new solutions resulting from these different methods together with each other are also obtained. Our results in this article confirm that the five different methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. [ABSTRACT FROM AUTHOR]

Several possible definitions for a multireference second-order perturbation theory are suggested. These are tested against some standard test problems from the literature. [ABSTRACT FROM AUTHOR]

Approximation theory for Lyapunov and Sacker-Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge-Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

The main goal of controller design in teleoperation systems is to achieve optimal performance, transparency and stability in presence of factors such as time delay in communication channel and modeling uncertainties. The teleoperation systems usually have complex dynamic. Consequently, differential equation solution of optimal control problem is difficult and complex for them. This paper presents a novel method for designing optimal controller based on singular perturbation framework for these systems. Firstly, we use the Taylor expansion to model the time delay, with considering time delay term; we derive a singular perturbation formulation for the teleoperation system. Using singular perturbation model and Chang decoupling transformation, singularly perturbed differential equations of optimal control problem is decomposed into the reduced order slow and fast differential equations. A formula is obtained that produces the solution of original differential equations of optimal control problem in terms of solutions of the slow and fast reduced order matrix differential equations. The reduced-order differential equations decrease the complexity of the optimal control problem for teleoperation systems. The simulations verify the effectiveness of the proposed control method and excellent performances tracking with high speed and small control signal. [ABSTRACT FROM AUTHOR]

This paper deals with the optimality system of an optimal control problem governed by a nonlinear elliptic inclusion and a nonsmooth cost functional. The system describing the state consists of a variational-hemivariational inequality, the solution mapping of which with respect to the control is proved to be weakly closed. Existence of optimal pairs for the optimal control problem is obtained. Approximation results and abstract necessary optimality conditions of first order are derived based on the adapted penalty method and nonsmooth analysis techniques. Moreover, the optimality system for a class of obstacle problems with nonmonotone perturbation is given. [ABSTRACT FROM AUTHOR]

We describe applications of asymptotic methods to problems of mathematical physics and mechanics, primarily, to the solution of nonlinear singularly perturbed problems in local domains. We also discuss applications of Padé approximations for transformation of asymptotic expansions to rational or quasi-fractional functions. [ABSTRACT FROM AUTHOR]

The problem of variational data assimilation for an evolution model is considered with the aim to identify the initial condition. The solvability of the optimality system is studied. Based on the adjoint equations, iterative algorithms for solving the problem are developed and justified. [ABSTRACT FROM AUTHOR]

In this paper, some new absolute perturbation bounds of partitioned generalized Hermitian positive definite eigenvalue problem are established by two different ways. Numerical results show that our bounds are sharper than the ones in the literature. [ABSTRACT FROM AUTHOR]